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Revision as of 22:55, 28 December 2024 editGregariousMadness (talk | contribs)Extended confirmed users1,309 edits ←Created page with '{{short description|A mathematical operator used in theoretical physics and topology}} '''Yang-Baxter operators''' are invertible linear endomorphisms with applications in theoretical physics and topology. These operators are particularly notable for providing solutions to the quantum Yang-Baxter equation, which originated in statistical mechanics, and for their use in constructing invariants of knots, links, and thre...'  Latest revision as of 02:27, 8 January 2025 edit undoMathKeduor7 (talk | contribs)Extended confirmed users1,455 editsm →Definition: ce 
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{{short description|A mathematical operator used in theoretical physics and topology}} {{short description|A mathematical operator used in theoretical physics and topology}}
'''Yang-Baxter operators''' are ] ] ] with applications in ] and ]. These operators are particularly notable for providing solutions to the quantum ], which originated in statistical mechanics, and for their use in constructing invariants of knots, links, and three-dimensional manifolds. '''Yang–Baxter operators''' are ] ] ] with applications in ] and ]. They are named after ] ] and ]. These ] are particularly notable for providing solutions to the quantum ], which originated in ], and for their use in constructing ] of ], links, and three-dimensional ].<ref name="Baxter1982">Baxter, R. (1982). "Exactly solved models in statistical mechanics". Academic Press. ISBN 978-0-12-083180-7.</ref><ref name="Yang1967">Yang, C.N. (1967). "Some exact results for the many-body problem in one dimension with repulsive delta-function interaction". '']''. 19: 1312–1315.</ref><ref name="Kauffman1991">Kauffman, L.H. (1991). "Knots and physics". Series on Knots and Everything. 1. World Scientific. ISBN 978-981-02-0332-1.</ref>


== Definition == == Definition ==
In the ] over a ] <math>k</math>, Yang-Baxter operators are <math>k</math>-linear mappings <math>R: V \otimes_k V \rightarrow V \otimes_k V</math>. The operator <math>R</math> satisfies the ''quantum Yang-Baxter equation'' if In the ] over a ] <math>k</math>, Yang–Baxter operators are ] <math>R: V \otimes_k V \rightarrow V \otimes_k V</math>. The operator <math>R</math> satisfies the ''quantum Yang-Baxter equation'' if
<blockquote><math>R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}</math></blockquote> <blockquote><math>R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}</math></blockquote>
where where
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The <math>\tau_{U,V}</math> represents the "twist" mapping defined for <math>k</math>-modules <math>U</math> and <math>V</math> by <math>\tau_{U,V}(u \otimes v) = v \otimes u</math> for all <math>u \in U</math> and <math>v \in V</math>. The <math>\tau_{U,V}</math> represents the "twist" mapping defined for <math>k</math>-modules <math>U</math> and <math>V</math> by <math>\tau_{U,V}(u \otimes v) = v \otimes u</math> for all <math>u \in U</math> and <math>v \in V</math>.


An important relationship exists between the quantum Yang-Baxter equation and the ]. If <math>R</math> satisfies the quantum Yang-Baxter equation, then <math>B = \tau_{V,V}R</math> satisfies <math>B_{12}B_{23}B_{12} = B_{23}B_{12}B_{23}</math>. An important relationship exists between the quantum Yang-Baxter equation and the ]. If <math>R</math> satisfies the quantum Yang-Baxter equation, then <math>B = \tau_{V,V}R</math> satisfies <math>B_{12}B_{23}B_{12} = B_{23}B_{12}B_{23}</math>.<ref name="Joyal1993">Joyal, A.; Street, R. (1993). "Braided tensor categories". '']''. 102: 20–78.</ref>

== Applications ==
Yang–Baxter operators have applications in ] and ].<ref name="Zamolodchikov1975">Zamolodchikov, A.B.; Zamolodchikov, A.B. (1975). "Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models". '']''. 120: 253–291.</ref><ref name="Jimbo1985">Jimbo, M. (1985). "A q-difference analogue of U(g) and the Yang-Baxter equation". '']''. 10: 63–69.</ref><ref name="Reshetikhin1991">Reshetikhin, N.Yu.; Turaev, V.G. (1991). "Invariants of 3-manifolds via link polynomials and quantum groups". '']''. 103: 547–597.</ref>


== See also == == See also ==
* ] * ]
* ] * ]
* ]
* ]
* ]
* ]


== References == == References ==
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] ]
]

Latest revision as of 02:27, 8 January 2025

A mathematical operator used in theoretical physics and topology

Yang–Baxter operators are invertible linear endomorphisms with applications in theoretical physics and topology. They are named after theoretical physicists Yang Chen-Ning and Rodney Baxter. These operators are particularly notable for providing solutions to the quantum Yang–Baxter equation, which originated in statistical mechanics, and for their use in constructing invariants of knots, links, and three-dimensional manifolds.

Definition

In the category of left modules over a commutative ring k {\displaystyle k} , Yang–Baxter operators are k {\displaystyle k} -linear mappings R : V ⊗ k V → V ⊗ k V {\displaystyle R:V\otimes _{k}V\rightarrow V\otimes _{k}V} . The operator R {\displaystyle R} satisfies the quantum Yang-Baxter equation if

R 12 R 13 R 23 = R 23 R 13 R 12 {\displaystyle R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}}

where

R 12 = R ⊗ k 1 {\displaystyle R_{12}=R\otimes _{k}1} ,
R 23 = 1 ⊗ k R {\displaystyle R_{23}=1\otimes _{k}R} ,
R 13 = ( 1 ⊗ k τ V , V ) ( R ⊗ k 1 ) ( 1 ⊗ k τ V , V ) {\displaystyle R_{13}=(1\otimes _{k}\tau _{V,V})(R\otimes _{k}1)(1\otimes _{k}\tau _{V,V})}

The τ U , V {\displaystyle \tau _{U,V}} represents the "twist" mapping defined for k {\displaystyle k} -modules U {\displaystyle U} and V {\displaystyle V} by τ U , V ( u ⊗ v ) = v ⊗ u {\displaystyle \tau _{U,V}(u\otimes v)=v\otimes u} for all u ∈ U {\displaystyle u\in U} and v ∈ V {\displaystyle v\in V} .

An important relationship exists between the quantum Yang-Baxter equation and the braid equation. If R {\displaystyle R} satisfies the quantum Yang-Baxter equation, then B = τ V , V R {\displaystyle B=\tau _{V,V}R} satisfies B 12 B 23 B 12 = B 23 B 12 B 23 {\displaystyle B_{12}B_{23}B_{12}=B_{23}B_{12}B_{23}} .

Applications

Yang–Baxter operators have applications in statistical mechanics and topology.

See also

References

  1. Baxter, R. (1982). "Exactly solved models in statistical mechanics". Academic Press. ISBN 978-0-12-083180-7.
  2. Yang, C.N. (1967). "Some exact results for the many-body problem in one dimension with repulsive delta-function interaction". Physical Review Letters. 19: 1312–1315.
  3. Kauffman, L.H. (1991). "Knots and physics". Series on Knots and Everything. 1. World Scientific. ISBN 978-981-02-0332-1.
  4. Joyal, A.; Street, R. (1993). "Braided tensor categories". Advances in Mathematics. 102: 20–78.
  5. Zamolodchikov, A.B.; Zamolodchikov, A.B. (1975). "Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models". Annals of Physics. 120: 253–291.
  6. Jimbo, M. (1985). "A q-difference analogue of U(g) and the Yang-Baxter equation". Letters in Mathematical Physics. 10: 63–69.
  7. Reshetikhin, N.Yu.; Turaev, V.G. (1991). "Invariants of 3-manifolds via link polynomials and quantum groups". Inventiones Mathematicae. 103: 547–597.
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